A357479
a(n) = (n!/6) * Sum_{k=0..n-3} 1/k!.
Original entry on oeis.org
0, 0, 0, 1, 8, 50, 320, 2275, 18256, 164388, 1644000, 18084165, 217010200, 2821132886, 39495860768, 592437911975, 9479006592160, 161143112067400, 2900576017214016, 55110944327067273, 1102218886541346600, 23146596617368279930, 509225125582102160000
Offset: 0
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Table[n!/6 Sum[1/k!,{k,0,n-3}],{n,0,30}] (* Harvey P. Dale, Apr 02 2023 *)
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a(n) = n!/6*sum(k=0, n-3, 1/k!);
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a(n) = n!*sum(k=0, n, binomial(k, 3)/k!);
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my(N=30, x='x+O('x^N)); concat([0, 0, 0], Vec(serlaplace(x^3/6*exp(x)/(1-x))))
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my(N=30, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=3, N, k!*x^k/(1-x)^(k+1))/6))
A357480
a(n) = (n!/24) * Sum_{k=0..n-4} 1/k!.
Original entry on oeis.org
0, 0, 0, 0, 1, 10, 75, 560, 4550, 41076, 410970, 4521000, 54252495, 705283150, 9873965101, 148109477880, 2369751647900, 40285778016680, 725144004303300, 13777736081766576, 275554721635336365, 5786649154342069650, 127306281395525539615, 2928044472097087420000
Offset: 0
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a(n) = n!/24*sum(k=0, n-4, 1/k!);
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a(n) = n!*sum(k=0, n, binomial(k, 4)/k!);
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my(N=30, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(serlaplace(x^4/24*exp(x)/(1-x))))
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my(N=30, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=4, N, k!*x^k/(1-x)^(k+1))/24))
A266083
a(n) = Sum_{k = 0..n - 1} (a(n - 1) + k) for n>0, a(0) = 1.
Original entry on oeis.org
1, 1, 3, 12, 54, 280, 1695, 11886, 95116, 856080, 8560845, 94169350, 1130032266, 14690419536, 205665873595, 3084988104030, 49359809664600, 839116764298336, 15104101757370201, 286977933390033990, 5739558667800679990, 120530732023814280000, 2651676104523914160231
Offset: 0
a(0) = 1;
a(1) = 1 + 0 = 1;
a(2) = 1 + 0 + 1 + 1 = 3;
a(3) = 3 + 0 + 3 + 1 + 3 + 2 = 12;
a(4) = 12 + 0 + 12 + 1 + 12 + 2 + 12 + 3 = 54;
a(5) = 54 + 0 + 54 + 1 + 54 + 2 + 54 + 3 + 54 + 4 = 280, etc.
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Table[(2 n! + Exp[1] n (n - 1) Gamma[n - 1, 1])/2, {n, 0, 22}]
RecurrenceTable[{a[n] == n*a[n - 1] + Binomial[n, 2], a[0] == 1}, a, {n, 0, 20}] (* G. C. Greubel, Dec 22 2015 *)
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a(n) = (2*n! + exp(1)*n*(n-1)*incgam(n-1, 1))\/2
A360877
Array read by antidiagonals: T(m,n) is the number of (undirected) paths in the rook graph K_m X K_n.
Original entry on oeis.org
0, 1, 1, 6, 12, 6, 30, 129, 129, 30, 160, 1984, 4536, 1984, 160, 975, 45945, 310542, 310542, 45945, 975, 6846, 1524156, 38298270, 111933456, 38298270, 1524156, 6846
Offset: 1
Array begins:
==============================================
m\n| 1 2 3 4 5 ...
---+------------------------------------------
1 | 0 1 6 30 160 ...
2 | 1 12 129 1984 45945 ...
3 | 6 129 4536 310542 38298270 ...
4 | 30 1984 310542 111933456 ...
5 | 160 45945 38298270 ...
...
- Eric Weisstein's World of Mathematics, Graph Path.
- Eric Weisstein's World of Mathematics, Rook Graph.
A365617
Iterated Pochhammer symbol.
Original entry on oeis.org
1, 1, 2, 24, 421200, 13257209623458438290962108800
Offset: 0
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a:= proc(n) option remember; `if`(n<0, 0,
(z-> mul(z+j, j=0..n-1))(a(n-1)))
end:
seq(a(n), n=0..5); # Alois P. Heinz, Sep 12 2023
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FoldList[Pochhammer, 1, Range[5]] (* Amiram Eldar, Sep 12 2023 *)
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P(x, y) = my(P=1); for (i=0, y-1, P *= x+i); P;
a(n) = my(x=1); n--; for (i=1, n, x = P(x, i+1)); x; \\ Michel Marcus, Sep 13 2023
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from gmpy2 import *
from functools import reduce
gamma = lambda n: fac(n - 1)
Pochhammer = lambda z,n: gamma(n + z) // gamma(z)
list_Pochhammer = lambda lst: int(reduce((lambda x, y: Pochhammer(x, y)), lst)) if len(lst) > 0 else 1
print([list_Pochhammer(range(1, n + 1)) for n in range(0, 6)])
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from functools import reduce
from sympy import rf
def A365617(n): return reduce(rf,range(1,n+1),1) # Chai Wah Wu, Sep 15 2023